Robust discrete-time super-hedging strategies under AIP condition and under price uncertainty

We solve the problem of super-hedging European or Asian options for discrete-time financial market models where executable prices are uncertain. The risky asset prices are not described by single-valued processes but measurable selections of random sets that allows to consider a large variety of models including bid-ask models with order books, but also models with a delay in the execution of the orders. We provide a numerical procedure to compute the infimum price under a weak no-arbitrage condition, the so-called AIP condition, under which the prices of the non negative European options are non negative. This condition is weaker than the existence of a risk-neutral martingale measure but it is sufficient to numerically solve the super-hedging problem. We illustrate our method by a numerical example.


Introduction
As observed in practice, the executed value of an asset may depend on the order sent by the trader and, also, on the quantities available in the order book. Among the possible causes of the well-known slippage phenomenon, delays in the execution of the orders, liquidity disorders, market impacts, or transaction costs may influence the executed value. An approach to overcome this difficulty is to assume that we do not know in advance the traded prices. In that case, as proposed in the paper, the order that the trader sends is a mapping that associates to each possible price available in the market a quantity to sell or buy. This is exactly what we generally observe in practice, in a presence of an order book for example, since there is no single price.
On the contrary, it is traditional in mathematical finance to suppose that we first observe a (new) single market price and, then, we choose almost instantaneously the number of assets to sell or buy in order to revise the portfolio. This means that the last traded price is kept constant long enough in the order book. Moreover, it coincides with a bid and ask price so that the buy and sell orders are executed at the same value.
In real life, there may be delayed information, see the recent paper [1] or [24,30] among others on stochastic control. The delayed information in the problem of pricing is sometimes modeled through incomplete or restricted information as in [14,18,19,29] or using a two filtrations setting as in [13].
Another type of uncertainty is due to the choice of the model supposed to approximate the real financial market [5]. Model risk may lead to price misevaluations that are studied in recent papers, in the growing field of robust finance. Since the seminal work of Knight [20], it is now broadly accepted that uncertainty may be described by a parametrized family of models, instead of considering only one model, if there is a lack of information on the parameters, see [3,4,9,16,22,25,31]. Other models consider that the market is driven by a family of probability measures in such a way that uncertainty stems from the existence of several possible reference probability measures determining which events are negligible, see [6,7,8,11,12,17,23,26].
In any case, uncertainty is taken into account in the literature by considering either several probabilistic structures, e.g. a family of reference probability measures and filtrations for the same price process or a family of price process models on the same stochastic basis. In the recent paper [27], the choice is made to fix only one filtered probability space on which a collection of stochastic processes describes the possible dynamics of the stock prices. We follow this alternative approach. Precisely, we consider a unique stochastic basis but we suppose that, in discrete time, the next stock prices at any time are not modeled by a unique vector-valued random variable as it is usual to do. Instead, we assume that the next stock prices belong to a collection of possible processes. The approach we adopt in our paper is slightly different from [27] in the sense that the collections of possible prices we consider are connected from time to time in such a way that it is possible to represent them through measurable random sets.
Moreover, a less common type of uncertainty is introduced in this paper. Recall that it is usual in the literature, even in the recent papers on robust finance, to suppose that the transactions are executed at a price which is known in advance. For example, in the Black and Scholes model, the delta-hedging strategy for the European Call option at time t is a function Φ(t, S t ) of the single price S t observed at time t. In practice, the strategy is discretized at some dates (t i ) i=0,...,n with n → +∞ so that the number of stocks to trade at time t i is ∆Φ t i = Φ(t i , S t i ) − Φ(t i−1 , S t i−1 ). In the case where ∆Φ t i < 0, the executed price at time t i should be a bid price in the order book and an ask price otherwise, i.e. there should be at least two possible prices.
We take into account this ambiguity or uncertainty in our paper by assuming that there may be several possible executable prices at the next instant. This means in particular that we do not know in advance the price when we send an order to be executed. Precisely, an executed price S t at time t is only F t+1 -measurable where F t describes the market information available at time t. This is illustrated in our numerical example where the stock price is modeled by a pair of bid and ask prices.
This article addresses the super-hedging problem of European or Asian options under uncertainty and may be easily adapted to American options in discrete time. Here the uncertainty mainly refers to the uncertainty in executed prices due to the delay, which is modeled by random sets, and there is one single physical probability measure. Moreover, uncertainty may also refers to the presence of an order book so that several prices may exist and depend on the traded volumes. The advantage of the approach we consider is its flexibility, including a large variety of possible models, e.g. with transaction costs or limit order books. Contrarily to the classical approach, we do not suppose the existence of a risk-neutral probability measure but we work under the AIP condition of [2,10], i.e. we suppose that the super-hedging prices of the nonnegative European claims are non-negative, as it is easily observed in the real financial market. We recall that the AIP condition is weaker than the usual NA condition but it is sufficient to deduce numerically tractable pricing estimations, as illustrated in our numerical example.
The paper first focuses on the one-period case, see Section 3.1, and the multi-period case is automatically obtained by (measurably) paste all periods together. The one-period hedging problem can be described as: Here S t is a possible executed price which is F t+1 -measurable, θ t−1 is a trading strategy which is made at time t − 1 and its outcome is revealed at the same time t as S t−1 due to execution delay and, thus, V t−1 , which models the portfolio value at time t − 1, is also F t -measurable; g is an Asian option to be hedged while Λ t ((S u ) u≤t−1 ) represents the set of all possible prices S t that can be traded strictly after time t. The problem is essentially converted to the one without delay by taking supremum conditioned on F t in the above equation, and the (minimal) super-hedging price is provided in Theorem 3.1 in terms of the concave envelope of some related function restricted on the conditional closure of Λ t ((S u ) u≤t−1 ), see [15]. Properties of the hedging price, including continuity, convexity, and measurability are analyzed in Section 3.2. These properties are important to deduce backwardly the multi-period case which involves a measurable pasting.
The benefit of our approach is its easy implementation as illustrated in Section 4. Indeed, roughly speaking, our main results state that we only need to know the range of the future price values in terms of the observed prices to deduce the strategy θ t to be followed. This can be achieved from a historical data. The strategy depends at time t on the price S t , i.e. θ t = θ t (S t ) where S t is only revealed at time t + 1 so that the order a time t is the F t -measurable mapping z → θ t (z) and not θ t (S t ). Note that the executed price S t will depend on the model, e.g. S t may be one of the several bid and ask prices, and the delayed observation of S t at time t + 1 allows to deduce the quantity θ t (S t ) to hold in the portfolio.

Formulation of the problem
Let (Ω, (F t ) t∈{0,...,T +1} , F T , P ) be a filtered complete probability space where T is the time horizon. We suppose that F 0 is the trivial σ-algebra and the σ-algebra F t represents the information available on the market at time t. The financial market we consider is composed of d risky assets and a bond S 0 . We assume without loss of generality that S 0 = 1.
In the following, we shall consider random subsets A of R d , i.e. A = A(ω) may depend on ω ∈ Ω. We then denote by L 0 (A, F t ) the set of all random variables X t which are F t -measurable and satisfies Let us consider, for each t ≤ T + 1, Λ t ⊆ L 0 (R d + , F t+1 ) a collection of F t+1 -measurable random variables representing the possible executable prices for the risky assets between time t and time t + 1. We suppose that, at time t, the set Λ t may depend on the observed traded prices before time t, i.e. to each vector of prices (S u ) u≤t−1 , we associate a set Λ t = Λ t ((S u ) u≤t−1 ) representing the possible next prices S t after time t given that we have observed the executed prices (S u ) u≤t−1 at time t. We adopt the financial principle that the executed price S t is only known strictly after the order is sent at time t but before time t + 1.
Recall that S t represents the prices (S 1 t , . . . , S d t ) of d ≥ 1 risky assets proposed by the market to the portfolio manager when selling or buying. A typical case could be Λ t = L 0 (I t , F t+1 ) with where (S bj ) j=1,...,d and (S aj ) j=1,...,d are respectively the bid and the ask price processes observed in the market between time t and t + 1 that may depend on (S u ) u≤t−1 . They are not necessary the best bid/ask prices as, in practice, the real transaction price may be a convex combination of bid and ask prices. Indeed, a transaction is generally the result of an agreement between sellers and buyers but it also depends on the traded volume. Clearly, the portfolio manager does not benefit in general from the last traded price observed in the market when sending an order. On the contrary, he should face an uncertain price S t that depends on the type of order (and may be not executed) but it also depends on some random events he does not control, e.g. slippage.
A simple way to model this phenomenon is to suppose that the executable prices obtained by the manager belong to random intervals.
Example 2.3. Another interesting case is when Λ t = {S θ t : θ ∈ Θ} is a parametrized family of random variables. For instance, consider fixed processes (ξ u ) u≤T and (m u ) u≤T adapted to (F t+1 ) t=0,...,T and independent of F t . Let C be a compact subset of R and suppose that S −1 is given. We define recursively In this model, there is an uncertainty on prices because of the unknown parameter (volatility) σ. This is a classical problem in robust finance, see for example [22].
A portfolio strategy is an (F t+1 ) t=−1,...,T -adapted process θ = (θ 0 , θ) where, for all t = 0, . . . , T , θ t ∈ R d (resp. θ 0 t ∈ R) describes the quantities of risky assets (resp. the bond) held in the portfolio between time t and time t + 1. Since the strategies are not supposed to be adapted to (F t ) t=0,...,T but only adapted to (F t+1 ) t=0,...,T , the manager is not supposed to control the quantity of assets he wants to sell or buy. This is what happens in practice because the orders are not necessarily executed, for instance in the case of limit stock market orders. Precisely, the portfolio manager may send an F t -measurable order at time t that depends on the uncertain price S t which is only F t+1 measurable. For instance, such an order could be Buy at most 1000 units at a price less than or equal to 145 euros so that the strategies and the executed prices are linked. In the example, the executed quantity should be deduced from an order book as the minimum between 1000 and the number of assets we may obtain for a price less that 145. Then, the executed price is a weighted average of all prices available for less than 145 in the order book.
For such a strategy θ = (θ 0 , θ), we define the portfolio process with initial endowment V 0 ∈ L 0 (R, F 1 ), as the liquidation value Recall that S t is observed strictly after the portfolio manager sends an order for θ t at time t. In the super-hedging problem we solve, we expect orders which are mapping Here the notation xy is used to designate the Euler scalar product between two vectors x, y of R d .
In the following, we only consider self-financing portfolio processes Vθ, i.e. they satisfy by definition: This means that the cost of the new portfolio allocation (θ 0 t , θ t ), i.e. buying or selling the quantities ( , at the executed price S t is charged to the cash account. Therefore, The aim of the paper is to solve the following problem: Construct the minimal super-hedging strategy of an Asian option whose payoff is g(S 0 , . . . , S T ) for some convex deterministic function g on (R d ) T +1 . Because of price uncertainty, this means that we shall construct a self-financing strategy θ and we shall determine the minimal initial endowment are for t ≤ T . Note that V t is F t+1 -measurable hence one more step is necessary to deduce the initial endowment P 0 at time t = 0 we need for initiating a super-hedging portfolio process V , i.e. P 0 ≥ V 0 . Indeed, P 0 should be F 0 -measurable, i.e. a constant, or equivalently P 0 ≥ ess sup F 0 (V 0 ). We refer to [10] for the definitions of conditional essential supremum and infimum.
3. The super-hedging problem

The one time step resolution
We first introduce the basic tools and theoretical results we need in this section. A set Λ of measurable random variables is said F-decomposable if for any finite partition (F i ) i=1,...,n ⊆ F of Ω, and for every family In the following, we denote by Σ(Λ) the F-decomposable envelope of Λ, i.e. the smallest F-decomposable family containing Λ. Notice that We now introduce the general one step problem between the dates t − 1 and t for t ≥ 1. To do so, we suppose that after time t − 1 but strictly before time t the portfolio manager observes the price S t−1 , as a consequence of their order, see Definition 2.1. More precisely, the portfolio manager knows (S u ) u≤t−2 at time t − 1 and sends an order at time t − 1 which is executed with a delay so that the executed price In the following, we consider the σ-algebra F t = σ(S u : u ≤ t − 1) for all t ≥ 1. Let us consider a random function g t defined on (R d ) t+1 , t ≥ 1. We assume that the mapping we also expect a dependence between θ t−1 and (S u ) u≤t−1 as we shall see later. Nevertheless, we do not suppose an explicit dependence of Λ t ((S u ) u≤t−1 ) with respect to θ t−1 , which is an open problem. We observe by lower-semicontinuity that (3.1) holds if and only if This means that we may suppose w.
In the formula above, cl(I t ((S u ) u≤t−1 )|F t ) is the conditional closure of I t ((S u ) u≤t−1 ), i.e. the smallest F t -measurable closed random set which contains I t ((S u ) u≤t−1 ) almost surely. We refer the readers to [15,Theorem 3.4] for the existence and uniqueness of such conditional random set. Moreover, and 0 otherwise. Notice that f * t−1 is convex and l.s.c. as a supremum (on cl(I t ((S u ) u≤t−1 )|F t )) of convex and l.s.c. functions. Moreover, by [15,Theorem 3.4 We deduce that the F t -measurable prices at time t − 1 are given by the Minkowski sum The second step is to determine the infimum super-hedging price as To do so, we use the arguments of [10, Theorem 2.8] and we obtain our first main result: 3) and the set of all prices given by (3.4). Then, the infimum price given by (3.5), Proof. This is a consequence of the following chain of equalities: Note that we do not need to suppose no-arbitrage conditions to establish the very general pricing formula above. It is only based on the lower-semicontinuity and measurability assumptions satisfied by the payoff g.

Main properties satisfied by the one time step infimum super-hedging price
The results of this section are the main contribution of our paper. They are needed to propagate the one time step pricing procedure of Section 3.1 to the multi-period case. In the following, we suppose that, for all price process (S u ) u≤t−1 , there exists α t−1 ∈ L 0 (R d , F t ) and β t−1 ∈ L 0 (R, F t ) that may depend on (S u ) u≤t−1 such that This is the case for Asian options whose payoffs are for example of the form k(S 0 + S 1 + · · · + S t − K) + , k ≥ 0. By [10][Theorem 2.8], we know that We first establish the following result: 2 . By the Hahn-Banach separation theorem and a measurable selection argument, there exists a non null Multiplying the inequality by a sufficiently large positive multiplier, we may suppose that . At last, suppose that z → g t (S 0 , S 1 , . . . , S t−1 , z) is bounded from below by m t−1 ∈ L 0 (R, F t ) on cl(I t |F t ) and S t−1 ∈ conv cl(I t |F t ). Then, S t−1 = lim n→∞ S n where S n ∈ conv cl(I t |F t ), i.e. S n = Jn i=1 λ i,n x i,n where λ i,n ≥ 0 with Jn i=1 λ i,n = 1 and x i,n ∈ cl(I t |F t ) for all i, n. Consider (α, β) such that αx + β ≥ g t (S 0 , . . . , S t−1 , x) for all x ∈ cl(I t |F t ). Then, αS t−1 + β = lim n→∞ (αS n + β) with

a.s. and convex or bounded from below by
In particular, the infimum super-hedging price of any non negative payoff function is finite if and only if it is non negative or equivalently if S t−1 ∈ conv cl(I t (S u ) u≤t−1 |F t ).
As studied in [10], the non negativity of the prices for the zero claim or more generally for non negative European call options corresponds to a weak no arbitrage condition (AIP) which is naturally observed in practice. Adapted to our setting, we introduce the following definition: Definition 3.4. We say that condition AIP holds between t − 1 and t if the prices at time t − 1 of the time t zero claim is non negative for every price process (S u ) u≤t−1 . Moreover, we say that the condition AIP holds when AIP holds at any time step.
As observed in [10] and above, when AIP fails, the infimum of the zero claim, and more generally of non negative payoffs, may be −∞. In that case, the numerical procedure we develop in this paper is still valid but unrealistic and non-implementable in practice. By Corollary 3.3, we have: for any price process (S u ) u≤t−1 , i.e.
In the following, if g is a function defined on R d and D is a subset of R d , we denote by conc(g, D) the (relative) concave envelope of g on D, i.e. the smallest concave function defined on R d which dominates g only on D. Observe that g ≤ h on D is equivalent to g − δ D ≤ h on R d . Therefore, conc(g, D) always exists as soon as g is dominated by an affine function on D.
The following result allows us to compute the infimum price rather easily.

almost surely. Consider the concave envelope
Then, Proof. By definition, h t−1 is the smallest concave function which dominates g. We deduce that the set of all affine functions dominating g coincides with the set of all affine functions dominating h t−1 . By (3.8) we deduce that (3.9) holds.

Proof. It is clear by Lemma 3.6 that
At last, applying (3.11) with x = S t ∈ I t ((S u ) u≤t−1 ) ⊆ cl(I t ((S u ) u≤t−1 )|F t ), we deduce that Since x → g t (S 0 , . . . , S t−1 , x) is l.s.c., we consider the following random set: where (γ n t ) n≥1 is a Castaing representation of conv cl(I t (S u ) u≤t−1 |F t ). Since G t is F t × B(R d )measurable and G t = ∅ a.s, it admits a measurable selection which is a measurable strategy θ t for the price p t−1 ((S u ) u≤t−1 ). Remark 3.8. As the function h t−1 in (3.10) is concave and finite a.s. on the conditional closure conv cl(I t (S u ) u≤t−1 |F t ), see proof of Proposition 3.2, the super-differential ∂h(S t−1 ) of h t−1 at the point S t−1 is not empty when S t−1 belongs to the interior of conv cl(I t (S u ) u≤t−1 |F t ).
The following result proves the measurability of the infimum super-hedging price p t−1 ((S u ) u≤t−1 ) with respect to (S u ) u≤t−1 . To do so, we suppose the existence of a Castaing representation, see [21,28].
The rest of this section aims to prove that, under some technical conditions, the mapping (S u ) u≤t−1 −→ p t−1 ((S u ) u≤t−1 ) is lower-semicontinuous, which is needed to propagate backwardly the numerical procedure of Theorem 3.5 in the multi-step model. Definition 3.10. We say that the mapping is lower-semicontinous if the following property holds: For all sequence of price processes ((S n u ) u≤t−1 ) n≥1 converging a.s. to a process (S u ) u≤t−1 , and for all z ∈ cl(I t ((S u ) u≤t−1 )|F t ), there exists a sequence (z n ) n≥1 such that lim n z n = z and z n ∈ cl(I t ((S n u ) u≤t−1 )|F t ) for all n ≥ 1.
In the following, we define the closed convex random sets where B(0, ) is the closed ball of center z = 0 and radius > 0. We say that the mapping z → E t−1 ((S u ) u≤t−1 , z) is convex if, for all α ∈ [0, 1], and z 1 , z 2 ∈ R d , we have Note that this convexity property above is automatically satisfied if d = 1.

Proposition 3.12. Consider a payoff function g t defined on
By assumption, we know that for all z ∈ cl(I t ((S u ) u≤t−1 )|F t ), there exists a sequence z n ∈ cl(I t ((S n u ) u≤t−1 )|F t ) such that lim n z n = z. We may suppose that |z − z n | ≤ where > 0 is arbitrarily fixed. By assumption, for all z ∈ cl(I t ((S n u ) u≤t−1 )|F t ) in the ball B(z, ) of center z and radius , we have: where h (n) is an arbitrary affine function satisfying h (n) ≥ g t ((S n u ) u≤t−1 , · ) on cl(I t ((S n u ) u≤t−1 )|F t ). Let us define By convention, we set inf ∅ = −∞. Let us show that h (n) is concave. To see it, observe that We only need to consider the case where E n (z 1 ) = ∅ and E n (z 2 ) = ∅. We deduce that E n (z) = ∅. Moreover, by assumption, any u ∈ E n (z) may be written as u = Taking the infimum in the left hand side of the inequality above, we deduce that h (n) ( for all h (n) . As S n t−1 ∈ E n (S t−1 ), for n large enough, under AIP, we deduce that Taking the infimum over all affine functions h (n) , we get that for n large enough: As is arbitrarily chosen, we may conclude that

Case of a convex payoff function
We shall prove that p t−1 ((S u ) u≤t−1 ) is a convex function of the price process (S u ) u≤t−1 if so Λ t−1 is. In the following, we say that the mapping is convex for the inclusion if, for λ ∈ [0, 1], for all price process (S u ) u≤t−1 , ( S u ) u≤t−1 .

Proposition 3.13. Suppose that the mapping
non negative and

z) is lower semi-continuous and convex almost surely
and suppose that the mapping Λ t−1 : Proof. Let (S u ) u≤t−1 , (S u ) u≤t−1 be two price processes. Let us define the following price process 1]. We consider the following random sets: By assumption, we have Λ t−1 ⊆ λΛ t−1 + (1 − λ) Λ t−1 for λ ∈ [0, 1]. Let h and h be two affine functions such that: λ) x. By above, we have:

Now, let us consider
Observe that αE Therefore, taking the infimum in the right side of the inequality above, we deduce that h is a (non negative) concave function with finite values. So, it is continuous and we have h(x) ≥ g t (x) for all x ∈ Λ t−1 . We deduce that Taking the infimum over all the affine functions h and h, we deduce that and the conclusion follows.
Remark 3.14. Suppose that the AIP condition holds and that (3.7) holds. Consider Moreover, the strategy is given by and g T is the payoff function.
At last, the order to be sent at time t is given by the deterministic mapping defined on R t by Remark 3.16 (Market impact). It is possible in our model to include a market impact. Indeed, it suffices to make the order (demand) mapping D t (x) = θ t (x) − θ t−1 (S t−1 ) coincided at time t with the supply mapping O t (x), i.e. the available quantity we may buy or sell at price x in the order book. By convention, O t is negative for bid prices and positive for ask prices. It is an increasing function on R + starting from O t (0+) = −∞ at price 0 (we can sell as many assets as we want to the market at price 0) and ending up with O t (+∞) = +∞, i.e. we can buy as many assets as we want to the market at price +∞. As soon as D t is bounded, there exists executable bid prices S b t in the order book such that The executed bid price is naturally the best one among all possible. Similarly, there exists executable ask prices S a t in the order book such that D t (S a t ) ≤ O t (S a t ) when D t (S a t ) ≥ 0 and the order may be executed at price S a t for the quantity D t (S a t ) ≤ O t (S a t ). Note that the executed bid price may be closed to 0 while the executed ask price may be very large. This liquidity phenomenon is then taken into account in the model through the conditional supports allowing to compute the strategy in our approach.

The multistep backward procedure
The main results of Section 3.2 for the one step model may be applied recursively, starting from time T , as the payoff function g T is known.
Consider the case where the conditional support cl(I t ((S u ) u≤t−1 )|F t ) admits a Castaing representation (ξ m ) m≥1 where ξ m = x m ((S u ) u≤t−1 ), for all m ≥ 1, and x m are Borel functions on (R d ) t . Then, by Proposition 3.9, we know that the infimum price at time T − 1 is a Borel function g T −1 of the prices S 0 , . . . , S T −1 . Then, we may repeat the procedure if we are in position to verify that g T −1 is also l.s.c. This is the case by Proposition 3.13 and Remark 3.14, under convexity conditions. Many questions could be investigated for future research, e.g. sensitivity to modeling assumptions, but also how to calibrate such a model from statistical estimations. Mainly, we need to estimate conditional supports. This is illustrated in the numerical example that we propose in the next section. A technical question is also to consider discontinuous payoff functions even if this is less usual in finance where g is generally a convex function. Actually, by Lemma 3.6, we may replace the payoff function by its concave envelope. Note that our analysis is general enough to consider a lot of models, e.g. with order books.

Formulation of the problem with d = 1
In this section we consider the example of the European call option at time T = 2, i.e. with the payoff function g(S 2 ) = (S 2 − K) + , K > 0. Let (S t ) t=0,1,2 be the executed price process. Recall that S t belongs to the random set Λ t , for t = 0, 1, 2, respectively. We suppose that the risk-free asset is given by S 0 = 1. Recall that there exist F t+1 -measurable closed random sets I t = I t ((S u ) u≤t−1 ) such that: We may suppose that Λ = Σ(Λ) so that S t ∈ I t a.s. for t = 0, 1, 2. At each step, we shall apply the procedure we have developed in the sections above. In particular, we seek for the strategy θ and we deduce the portfolio value V associated to the executed price process S. Then, we may estimate the error between the terminal value of V 2 and the payoff g 2 (S 2 ) that we denote by 2 = V 2 − g 2 (S 2 ). We start from a known price S −1 at time t = 0, which corresponds to the last traded price. We suppose that We make this choice for simplicity and that corresponds to the case where the bid and ask prices of the market coincide with the mid price S 0 . The order we sent is of the form buy or sell the quantity θ 0 (z) at the price z.
At time t = 1, we choose to model bid and ask prices S bid 1 , S ask  Note that the mapping s 1 → ∆θ 1 (s 1 ) is the F 1 -measurable order we send at time t = 1, see Figure 4.1. The later depends on S 0 , which is F 1 -measurable.

Explicit computation of the strategy
We deduce the portfolio value and the strategy value at any time by dominating the payoff function by the smallest affine function on the conditional support of S, as mentioned in (3.8). We consider the terminal payoff function g(S T ) = (S T − K) + for several strikes. Recall that S 2 ∈ Λ 2 (S 1 ) ∼ I 2 = [S 1 m 2 , S 1 M 2 ]. In order to compute the strategy θ 1 = θ 1 (S 1 ) we first compute the function ϕ 1 given by (3.8) which dominates the pay-off function g 2 on the conditional support cl(I 2 (S 1 )|F 2 ) = [S 1 m − 2 , S 1 M + 2 ].

Empirical results
For an observed price S −1 at time t = 0 (which corresponds to the last traded price), and for different strike values K, we test the infimum super-hedging strategy by computing the relative error R from a data set of 10 6 simulated prices S t for t ∈ 0, 1, 2. To do so, we wrote a script in Python. The relative error is given by In the following  We observe that the executed prices depend on the strike K > 0, i.e. there is a market impact of the orders on the prices. Indeed, as expected, the orders we send depend on the payoff function. As K increases, the payoff decreases and, as expected, the option price V 0 decreases. The distribution of S 1 admits two regimes as seen in Figure 4.13 that correspond to the bid and ask prices.
Notice that the proportion of the portfolio value invested in the risky assets at time t = 1 decreases as the payoff decreases. We also observe that this proportion decreases (resp. increases) when the price S decreases (resp. increases) between time t = 0 and t = 1, i.e. when ∆S 1 < 0 (resp. ∆S 1 ≥ 0). At last, the empirical results obtained for the relative error confirm the efficiency of the super-hedging strategy, see